What is the quotient of the rational expression below?
just look at the picture

Answer: A) Yes, mold A is an exponential function that decreases faster than mold B, which is eventually an increasing quadratic function.

Step-by-step explanation:

To determine whether the number of spores in mold B will ever be larger than in mold A, we need to compare the growth patterns of the two functions. The function f(x) = 100(0.75)^(x-1) represents mold A, and it is an exponential function. Exponential functions decrease as the exponent increases. In this case, the base of the exponential function is 0.75, which is less than 1. Therefore, mold A is a decreasing exponential function. The function g(x) = 100(x-1)^2 represents mold B, and it is a quadratic function. Quadratic functions can have either a positive or negative leading coefficient. In this case, the coefficient is positive, and the function represents a parabola that opens upwards. Therefore, mold B is an increasing quadratic function. Since mold B is an increasing function and mold A is a decreasing function, there will be a point where the number of spores in mold B surpasses the number of spores in mold A. Thus, the correct answer is:

A) Yes, mold A is an exponential function that decreases faster than mold B, which is eventually an increasing quadratic function.

Answer: x=43

Step-by-step explanation:

Looks like the 2 angles are a linear pair, 2 angles that make up a line.  So if added they equal 180

Equation:

x + 7 + 3x + 1 = 180                   >Combine like terms

4x +8 = 180                               >Subtract 8 from both sides

4x = 172                                    >Divide both sides by 4

x = 43

Step-by-step explanation:

See image

The missing angle measures in parallelogram RSTU are:

The opposite angles of the parallelogram are the same.

From the diagram:

∠S = ∠U and ∠R = ∠T

Given:

Since the sum of angles in a quadrilateral is 360 degrees, hence:

[tex]\angle\text{R}+\angle\text{S}+\angle\text{T}+\angle\text{U} = 360[/tex]

Since ∠R = ∠T, then:

[tex]\angle\text{Y}+\angle\text{S}+\angle\text{T}+\angle\text{U} = 360[/tex]

[tex]2\angle\text{T} + 70+70 = 360[/tex]

[tex]2\angle\text{T} =360-140[/tex]

[tex]2\angle\text{T} = 220[/tex]

[tex]\angle\text{T} = \dfrac{220}{2}[/tex]

[tex]\bold{\angle T = 110^\circ}[/tex]

Since ∠T = ∠R, then ∠R = 110°

Hence, m∠R = 110°, m∠T = 110°, m∠U = 70°. Option B is correct.

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Using the method of undetermined coefficients, one solution of the given differential equation is y = A cos(8t) + B sin(8t) + C e²t, where A, B, and C are constants.

To find a particular solution using the method of undetermined coefficients, we assume a solution of the form y = A cos(8t) + B sin(8t) + C e²t, where A, B, and C are undetermined coefficients to be determined.

We differentiate y to find y' and substitute the expressions into the given differential equation − 4y' + 67y = 80e²¹ cos(8t) + 32e²¹ sin(8t) + 9e²t. By comparing the coefficients of the trigonometric and exponential terms on both sides of the equation, we can solve for A, B, and C.

After determining the values of A, B, and C, we substitute them back into the assumed solution y = A cos(8t) + B sin(8t) + C e²t. This gives us one particular solution of the differential equation.

It's important to note that the method of undetermined coefficients may not work in all cases, especially when the non-homogeneous term has a similar form to the complementary solution. In such cases, variations of parameters or other techniques may be required.

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The maximal rate of change is given by the magnitude of the gradient vector: ||∇f||. Here, F = [T, y³, 3] is the vector field, and dS is the outward-pointing vector normal to the surface S. Therefore, the answer for option b is Flux = ∬S F · dS

So, let's calculate the gradient vector (∇f) and evaluate it at the point [x₀, y₀, z₀].

∇f = [∂f/∂x, ∂f/∂y, ∂f/∂z]

The maximal rate of change is given by the magnitude of the gradient vector: ||∇f||.

(b) To calculate the flux of the vector field F(x, y, z) = [T, y³, 3] across the surface S, we can use the surface integral:

Flux = ∬S F · dS

Here, F = [T, y³, 3] is the vector field, and dS is the outward-pointing vector normal to the surface S.

(c) To evaluate the surface integral ∬S fyz dS over the surface S, we need the parametric equations of the surface S.

Therefore, the answer for option b is Flux = ∬S F · dS

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Two-sample t-test would be the hypothesis test based on the scenario created.

A statistical test called the two-sample t-test is used to compare the means of two different independent groups to see if there is a statistically significant difference between them. It is frequently applied when contrasting the means of two various treatment groups or populations. To establish the statistical significance of the test, a t-value is calculated and then compared to a critical value derived from the t-distribution.

The scenario provided are two different independent group, to see if there is statistically significant difference between them, two sample t-test will be used.

The following steps are taken when conducting two sample t-test;

1. Formulate the null and alternative hypothesis

2. Collect and organize the data

3. Check assumptions

4. Calculate the test statistic

5. Determine the critical value and calculate the p-value

6. Make a decision

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We can use a two-sample t-test to compare the two sample means.

The appropriate hypothesis test to determine whether the means of two populations differ significantly is the two-sample t-test.

The two-sample t-test is used to compare the means of two independent groups.

The hypothesis testing of the two independent means is performed using the following hypotheses:

H0: µ1 = µ2 (null hypothesis)

H1: µ1 ≠ µ2 (alternative hypothesis)

Here, µ1 and µ2 are the population means of two different populations and are unknown. We use sample means x1 and x2 to estimate the population means.

In this scenario, the sample sizes of the two populations are greater than 30.

Therefore, we can use a two-sample t-test to compare the two sample means.

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The least common multiple (LCM) of the polynomials x² - 32x - 10 and 2x + 10 is 2(x + 2)(x - 10)(x + 5).

To calculate the LCM, we need to find the polynomial that contains all the factors of both polynomials, while excluding any redundant factors.

Let's first factorize each polynomial to identify their prime factors:

x² - 32x - 10 = (x + 2)(x - 10)

2x + 10 = 2(x + 5)

Now, we can construct the LCM by including each prime factor once and raising them to the highest power found in either polynomial:

LCM = (x + 2)(x - 10)(2)(x + 5)

Simplifying the expression, we obtain:

LCM = 2(x + 2)(x - 10)(x + 5)

Therefore, the LCM of x² - 32x - 10 and 2x + 10 is 2(x + 2)(x - 10)(x + 5).

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Answer:

Step-by-step explanation:

the present value that should be invested now to accumulate $8400 in 9 years at 7% compounded quarterly is approximately $5035.40.

To find the present value of $8400 accumulated over 9 years at an interest rate of 7% compounded quarterly, we can use the present value formula for compound interest:

PV = FV / [tex](1 + r/n)^{(n*t)}[/tex]

Where:

PV = Present Value (the amount to be invested now)

FV = Future Value (the amount to be accumulated)

r = Annual interest rate (as a decimal)

n = Number of compounding periods per year

t = Number of years

In this case, we have:

FV = $8400

r = 7% = 0.07

n = 4 (compounded quarterly)

t = 9 years

Substituting these values into the formula, we have:

PV = $8400 / [tex](1 + 0.07/4)^{(4*9)}[/tex]

Calculating the present value using a calculator or spreadsheet software, we get:

PV ≈ $5035.40

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Applying tangent theorems, we have: 1. Perimeter = 68, 2. measure of the intercepted arc = 76°.

One of the tangent theorems states that two tangents that intersect to form an angle outside a circle are congruent, and they form a right angle with the radius of the circle.

1. Applying the tangent theorem, we have:

JA = JB = 14

AL = CL = 12

CK = BK = 8

Perimeter = JA + JB + CL + AL + CK + BK

= 14 + 14 + 12 + 12 + 8 + 8

= 68.

2. Since the radius of the circle forms a right angle with the tangents, therefore, one part of the central angle opposite the intercepted arc would be:

180 - 90 - (104)/2

= 38°

Measure of the intercepted arc = 2(38) = 76°

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Answer:

tan 58° = 11/x

Step-by-step explanation:

The two legs form the right angle of the triangle.

One leg is x, and the other leg is 11.

Look at the 58° angle. The leg with length x is next to the 58° angle, so the leg with length x is the "adjacent" leg. The leg with length 11 is opposite the 58° angle, so that leg is the "opposite" leg. For the 58° angle, adj = x, and opp = 11.

Now you need to remember the definitions of the sine, cosine, and tangent ratios for a right triangle.

sin A = opp/hyp

cos A = adj/hyp

tan A = opp/adj

The only ratio that involves the adjacent and opposite legs is the tangent.

Answer:

tan 58° = 11/x

For the value of x = 4, matrix A will have no inverse.

If a matrix A has no inverse, then its determinant equals zero. The determinant of matrix A is defined as follows:

|A| = 1(2x3 - 3x2) - 2(1x3 - 3x1) + 3(1x2 - 2x1)

we can simplify and solve for x as follows:|A| = 6x - 12 - 6x + 6 + 3x - 6 = 3x - 12

Therefore, we must have 3x - 12 = 0 for matrix A to have no inverse.

Hence, x = 4. That is the value of x for which the matrix A does not have an inverse.

For the value of x = 4, matrix A will have no inverse.

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Hans will have approximately $3944.88 in his savings account after six years, assuming he makes no withdrawals.

To calculate the amount Hans will have in his savings account after six years with compound interest, we can use the formula for compound interest:

A = P(1 + r/n)^(n*t)

Where:

A is the final amount

P is the principal amount (initial deposit)

r is the annual interest rate (in decimal form)

n is the number of times interest is compounded per year

t is the number of years

In this case, Hans deposited $3500, the interest rate is 2% (0.02 in decimal form), and the interest is compounded continuously.

Using the formula, we have:

A = 3500 * (1 + 0.02/1)^(1 * 6)

Since the interest is compounded continuously, we use n = 1.

A = 3500 * (1 + 0.02)^(6)

Now, we can calculate the final amount after six years:

A = 3500 * (1.02)^6

A ≈ 3500 * 1.126825

A ≈ 3944.87875

After rounding to the nearest cent, Hans will have approximately $3944.88 in his savings account after six years, assuming he makes no withdrawals.

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Let's calculate the total number of beads that Lacey has based on the given information.

Answer: 39 beads

Step-by-step explanation:

Lacey has 14 red beads.

She has 6 fewer yellow beads than red beads. This means that the number of yellow beads is 14 - 6 = 8.

She also has 3 more green beads than red beads. This means that the number of green beads is 14 + 3 = 17.

To find the total number of beads, we add up the number of red, yellow, and green beads: 14 + 8 + 17 = 39.

Therefore, Lacey has a total of 39 beads.

The expression 1/tan(x) + tan(x) is equal to cos(x) + sin(x). Therefore, option B. Sin(x)cos(x) is correct.

To simplify the expression 1/tan(x) + tan(x), we need to find a common denominator for the two terms.

Since tan(x) is equivalent to sin(x)/cos(x), we can rewrite the expression as:

1/tan(x) + tan(x) = 1/(sin(x)/cos(x)) + sin(x)/cos(x)

To simplify further, we can multiply the first term by cos(x)/cos(x) and the second term by sin(x)/sin(x):

1/(sin(x)/cos(x)) + sin(x)/cos(x) = cos(x)/sin(x) + sin(x)/cos(x)

Now, to find a common denominator, we multiply the first term by sin(x)/sin(x) and the second term by cos(x)/cos(x):

(cos(x)/sin(x))(sin(x)/sin(x)) + (sin(x)/cos(x))(cos(x)/cos(x)) = cos(x)sin(x)/sin(x) + sin(x)cos(x)/cos(x)

Simplifying the expression further, we get:

cos(x)sin(x)/sin(x) + sin(x)cos(x)/cos(x) = cos(x) + sin(x)

Therefore, the expression 1/tan(x) + tan(x) is equal to cos(x) + sin(x).

From the given choices, the best answer that matches the simplified expression is:

B. sin(x)cos(x)

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The probability of the monkey picking a red candy from any of the bottles is 0.75.

Let L, M, S be the events that the monkey chooses the largest, mid-size and small bottle respectively.P(R) be the probability that the monkey chooses a red candy from the chosen bottle.

P(B) be the probability that the monkey chooses a blue candy from the chosen bottle.

P(L) = 0.5 (Given)

P(M) = 0.4 (Given)

P(S) = 1 - P(L) - P(M) = 0.1 (Since there are only three bottles)

Now, P(R/L) = 8/10

P(B/L) = 2/10

P(R/M) = 5/12

P(B/M) = 7/12

P(R/S) = 4/6

P(B/S) = 2/6

Now, Let's find the probability of the monkey picking a red candy:

P(R) = P(L)P(R/L) + P(M)P(R/M) + P(S)P(R/S)

P(R) = 0.5 × 8/10 + 0.4 × 5/12 + 0.1 × 4/6

P(R) = 0.75

The probability of the monkey picking a red candy from any of the bottles is 0.75.

Therefore, the correct answer is 0.75.

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Local minimum: (7, 3); Saddle point: (-3, 3).  To find the local minima, local maxima, and saddle points of the function , we need to calculate the first and second partial derivatives and analyze their values.

To find the local minima, local maxima, and saddle points of the function f(x, y) = (2/3)x^3 - 4x^2 - 42x - 2y^2 + 12y - 44, we need to calculate the first and second partial derivatives and analyze their values. First, let's find the first partial derivatives:

f_x = 2x^2 - 8x - 42; f_y = -4y + 12.

Setting these derivatives equal to zero, we find the critical points:

2x^2 - 8x - 42 = 0

x^2 - 4x - 21 = 0

(x - 7)(x + 3) = 0;

-4y + 12 = 0

y = 3.

The critical points are (x, y) = (7, 3) and (x, y) = (-3, 3). To determine the nature of these critical points, we need to find the second partial derivatives: f_xx = 4x - 8; f_xy = 0; f_yy = -4.

Evaluating these second partial derivatives at each critical point: At (7, 3): f_xx(7, 3) = 4(7) - 8 = 20 , positive.

f_xy(7, 3) = 0 ---> zero. f_yy(7, 3) = -4. negative.

At (-3, 3): f_xx(-3, 3) = 4(-3) - 8 = -20. negative;

f_xy(-3, 3) = 0 ---> zero; f_yy(-3, 3) = -4 . negative.

Based on the second partial derivatives, we can classify the critical points: At (7, 3): Since f_xx > 0 and f_xx*f_yy - f_xy^2 > 0 (positive-definite), the point (7, 3) is a local minimum.

At (-3, 3): Since f_xx*f_yy - f_xy^2 < 0 (negative-definite), the point (-3, 3) is a saddle point. In summary: Local minimum: (7, 3); Saddle point: (-3, 3).

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For part (a), we have found that a = 18822 and b = 18982 satisfy a^2 ≡ b^2 (mod N), where N = 61063. By computing gcd(N, a - b), we can find a nontrivial factor of N.

In part (a), we are given N = 61063 and two congruences: 18822 ≡ 270 (mod 61063) and 18982 ≡ 60750 (mod 61063). We observe that 270 = 2 · 3^3 · 5 and 60750 = 2 · 3^5 · 5^3. These congruences imply that a^2 ≡ b^2 (mod N), where a = 18822 and b = 18982.

To find a nontrivial factor of N, we compute gcd(N, a - b). Subtracting b from a, we get 18822 - 18982 = -160. Taking the absolute value, we have |a - b| = 160. Now we calculate gcd(61063, 160) = 1. Since the gcd is not equal to 1, we have found a nontrivial factor of N.

Therefore, in part (a), the values of a and b satisfying a^2 ≡ b^2 (mod N) are a = 18822 and b = 18982. The gcd(N, a - b) is 160, which gives us a nontrivial factor of N.

For part (b), a similar process can be followed to find the values of a, b, and the nontrivial factor of N.

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The constant of proportionality is 1.6 km/cm, and the real-world distance corresponding to a route of 3.2 cm on the map would be 5.12 km.

a) The constant of proportionality between the map and the real world can be calculated by dividing the real-world distance by the corresponding distance on the map.

In this case, since 1 cm on the map corresponds to 1.6 km in the real world, the constant of proportionality would be 1.6 km/1 cm, which simplifies to 1.6 km/cm.

b) To convert the distance of 3.2 cm on the map to the real-world distance, we can multiply it by the constant of proportionality. So, 3.2 cm ˣ  1.6 km/cm = 5.12 km.

Therefore, a route that measures 3.2 cm on the map would have a length of 5.12 km in the real world.

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Answer:

To represent Sal's situation, we can use an inequality to express the minimum earnings he needs to meet his weekly target.

Let's denote:

- E as Sal's earnings per week (in dollars)

- R as Sal's hourly rate ($17.50)

- H as the number of hours Sal works per week

Since Sal earns an hourly wage of $17.50, we can calculate his weekly earnings as E = R * H. Sal needs to earn at least $525 per week, so we can write the following inequality:

E ≥ 525

Substituting E = R * H:

R * H ≥ 525

Using the given information that R = $17.50, the inequality becomes:

17.50 * H ≥ 525

Therefore, the inequality that best represents Sal's situation is 17.50H ≥ 525.

Answer:

m² - 4

Step-by-step explanation:

(m-2) (m+2)

= m² + 2m - 2m - 4

= m² - 4

Answer: 0.11 cents a message

Step-by-step explanation:

Total of texts: 40

Total cost: $4.40

4.40/40

= 0.11

The two internal bisectors and one external bisector of the angles of a triangle meet the opposite sides in three collinear points.

When we consider a triangle, each angle has an internal bisector and an external bisector.

The internal bisector of an angle divides the angle into two equal parts, while the external bisector extends outside the triangle and divides the angle into two supplementary angles.

To prove that the two internal bisectors and one external bisector of the angles of a triangle meet the opposite sides in three collinear points, we need to understand the concept of angle bisectors and their properties.

First, let's consider one of the internal bisectors. It divides the angle into two equal parts and intersects the opposite side.

Since both angles formed by the bisector are equal, the opposite sides of these angles are proportional according to the Angle Bisector Theorem.

Now, let's focus on the second internal bisector. It also divides its corresponding angle into two equal parts and intersects the opposite side. Similarly, the opposite sides of these angles are proportional.

Next, let's examine the external bisector. Unlike the internal bisectors, it extends outside the triangle. It divides the exterior angle into two supplementary angles, and its extension intersects the opposite side.

To understand why the three bisectors meet at collinear points, we observe that the opposite sides of the internal bisectors are proportional, and the opposite sides of the external bisector are also proportional to the sides of the triangle.

This implies that the three intersecting points lie on a straight line, as they satisfy the condition of collinearity.

In conclusion, the two internal bisectors and one external bisector of the angles of a triangle meet the opposite sides in three collinear points due to the proportional relationship between the opposite sides formed by these bisectors.

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Explanation cannot be summarized in one row as it requires multiple factors and considerations to determine the asymptotes of different functions.

In order to find the horizontal and vertical asymptotes of a function, we need to analyze its behavior as x approaches infinity or negative infinity.

In the given question, we are provided with multiple functions (a) to (h) and asked to find their asymptotes, if any exist.

To find the horizontal asymptote, we look at the highest degree term in the numerator and denominator.

If the degrees are equal, the horizontal asymptote is the ratio of their coefficients.

If the degree of the numerator is greater, there is no horizontal asymptote.

For vertical asymptotes, we examine the values of x that make the denominator zero.

These values represent vertical lines that the graph approaches but never crosses.

By analyzing the given functions based on these criteria, we can determine whether they have horizontal or vertical asymptotes, if any.

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To determine the profit or loss at the current unit, the information regarding costs and revenue associated with the unit must be considered.

To calculate the profit or loss at the current unit, the revenue generated by the unit must be subtracted from the total costs incurred. If the result is positive, it represents a profit, while a negative result indicates a loss.

The calculation involves considering various factors such as production costs, operational expenses, and the selling price of the unit. By subtracting the total costs from the revenue generated, the net financial outcome can be determined.

It's important to note that without specific cost and revenue figures, it's not possible to provide an exact profit or loss amount. However, by performing the necessary calculations using the available data, the profit or loss at the current unit can be determined accurately, rounded to two decimal points for precision.

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The given regressions analyze the relationship between wages and gender by considering the average hourly earnings for females and males in a population. The coefficients in the equations provide insights into the average wage differences between genders.

The given question asks us to consider three regressions that hold for the same population. The three regressions are as follows:

1. Wage = A0 + A1 * Female + Ui
2. Wage = B0 + B2 * Male + Vi
3. Wage = C1 * Female + C2 * Male + Ei

In these equations, "Wage" refers to average hourly earnings, "U," "V," and "E" are the error terms of the regressions, and "Female" is a variable that takes the value of 1 if the observation refers to a female and 0 if it refers to a male. Similarly, "Male" is a variable that takes the value of 1 if the observation refers to a male.

Let's break down these equations:

1. The first regression equation states that the wage is equal to A0 plus the product of A1 and the "Female" variable, added to an error term "Ui."

2. The second regression equation states that the wage is equal to B0 plus the product of B2 and the "Male" variable, added to an error term "Vi."

3. The third regression equation states that the wage is equal to the product of C1 and the "Female" variable, plus the product of C2 and the "Male" variable, added to an error term "Ei."

These regressions are used to analyze the relationship between wages and gender. By including the variables "Female" and "Male" in the equations, we can estimate the impact of gender on wages.

The coefficients A1, B2, and C1 represent the average difference in wages between females and males, while the coefficients A0, B0, and C2 represent the average wages for males when the respective gender variable is 0.

It's important to note that these equations are specific to the population being studied and the variables included in the analysis.

The error terms (Ui, Vi, and Ei) account for factors not included in the regressions that affect wages, such as education, experience, and other socioeconomic variables.

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It will take at least 25 weeks for you to save enough money to purchase the car, assuming you currently have $350 in the bank and save an additional $60 per week.

To determine the number of weeks it will take for you to save enough money to purchase the car, we can set up an equation and solve for the number of weeks.

Let's denote the number of weeks as "w".

Given that you currently have $350 in the bank and plan to save an additional $60 per week, the amount of money you will have after "w" weeks can be represented as:

350 + 60w

We want this amount to be at least $1800, so we can set up the following inequality:

350 + 60w ≥ 1800

To find the number of weeks, we need to solve this inequality for "w".

Subtracting 350 from both sides of the inequality, we have:

60w ≥ 1450

Dividing both sides of the inequality by 60, we get:

w ≥ 24.167

Since the number of weeks must be a whole number, we can round up to the nearest whole number. Thus, it will take you at least 25 weeks to save enough money to purchase the car.

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Using mathematical strong induction, we can prove that for all n ≥ 1, 1 < an < 1/(1-a), given 0 < a < 1.

To prove the given statement using mathematical strong induction, we first establish the base case. For n = 1, we have a1 = 1 + a. Since a < 1, it follows that a1 = 1 + a < 1 + 1 = 2. Additionally, since a > 0, we have a1 = 1 + a > 1, satisfying the condition 1 < a1.

Now, we assume that for all k ≥ 1, 1 < ak < 1/(1-a) holds true. This is the induction hypothesis.

Next, we need to prove that the statement holds for n = k+1. We have a_k+1 = 1/ak + a. Since 1 < ak < 1/(1-a) from the induction hypothesis, we can establish the following inequalities:

1/ak > 1/(1/(1-a)) = 1-a

a < 1

Adding these inequalities together, we get:

1/ak + a > 1-a + a = 1

Thus, we have 1 < a_k+1.

To prove a_k+1 < 1/(1-a), we can rewrite the inequality as:

1 - a_k+1 = 1 - (1/ak + a) = (ak - 1)/(ak * (1-a))

Since 1 < ak < 1/(1-a) from the induction hypothesis, it follows that (ak - 1)/(ak * (1-a)) < 0.

Therefore, we have a_k+1 < 1/(1-a), which completes the induction step.

By mathematical strong induction, we have proven that for all n ≥ 1, 1 < an < 1/(1-a), given 0 < a < 1.

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The value of x, y and z in the interior angles of the parallelogram is 38, 81 and 75.

A parallelogram is simply quadrilateral with two pairs of parallel sides.

Opposite angles of a parallelogram are equal.

Consecutive angles in a parallelogram are supplementary.

From the diagram, angle ( 3x - 6 ) is opposite angle 108 degrees.

Since opposite angles of a parallelogram are equal.

( 3x - 6 ) = 108

Solve for x:

3x - 6 = 108

3x = 108 + 6

3x = 114

x = 114/3

x = 38

Also, consecutive angles in a parallelogram are supplementary.

Hence:

108 + ( y - 9 ) = 180

y + 108 - 9 = 180

y + 99 = 180

y = 180 - 99

y = 81

And

108 + ( z - 3 ) = 180

z + 108 - 3 = 180

z + 105 = 180

z = 180 - 105

z = 75

Therefore, the value of z is 75.

Learn more about parallelogram here: https://brainly.com/question/32441125

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